Guided waves result from multiple reflections in layered media. A guided wave occurs when a wave propagating in a layer is incident on a boundary of the layer at an angle to the surface normal greater than the “critical angle”. As is well known, when the incident angle to the surface normal is greater than the critical angle, the sine of the angle of transmission to the surface normal, as determined by Snell's Law, is greater than 1. A wave transmitted out of the layer may exist only as an evanescent wave, and the phenomenon of “total internal reflection” occurs. The waves are therefore trapped in the layer and propagate as guided waves within the layer that do not decay significantly with travel distance. A layer that transmit guided waves in this way is called a “waveguide”. The waveguide effect is shown schematically in FIG. 1. A wave 1 propagating in a layer 2 is incident on a boundary 3 of the layer. When the incident angle to the surface normal is greater than the critical angle θc, total internal reflection occurs leading to a guided wave. This is shown by the full line in FIG. 1. (If the angle of incidence, to the normal to the surface 3, is less than the critical angle then the wave is partially transmitted and partially reflected, as shown by the dotted lines.)
When the guided waves are recorded along the waveguide surface the apparent slowness or phase velocity of the guided waves exhibit dispersion—that is, they are each a function of the frequency of the waves. This dispersive character of guided waves can be used to obtain information about a waveguide or, more generally, about material properties of, and wave propagation velocities in, layered media. This is commonly achieved by solving an inverse problem, and matching a dispersion curve obtained for a numerical model of a waveguide to an observed dispersion curve. One such method is described by M. Roth and K. Holliger, in “Joint inversion of Rayleigh and guided waves in high-resolution seismic data using a genetic algorithm”, Society of Exploration Geophysicists, Expanded Abstracts, pp 1570–1573 (1998). They suggested picking the dispersion curves of Rayleigh waves and guided waves in dispersion images and inverting them to obtain the velocity of P-waves, the velocity of S-waves, and the density of the waveguide using a genetic algorithm. This method requires an iterative inversion for P-velocity, S-velocity and waveguide density in the top layer and halfspace—6 parameters in total—and so requires considerable computing power. Furthermore, this method requires the waveguide density, either as prior knowledge or as a parameter in the inversion.
Dispersion images have also been used, by J. Xia et al. in “Estimation of near-surface shear-wave velocity by inversion of Rayleigh waves”, Geophysics, Vol. 64, pp 691–700 (1999), to analyse dispersion curves of Rayleigh waves. The inversion of Rayleigh waves provides only S-velocities. The P-velocities are assumed to be known. This method also requires knowledge of the density of the waveguide layer.